Learning from the uncertain:
Modelling and forecasting of
infectious disease outbreaks


Sebastian Funk
16 November, 2017
Probabilistic programming meeting, Uppsala

Metcalf & Lessler (2017)

Forecasting the Ebola epidemic

Summer 2014

\(y=ax+b\)

\begin{eqnarray} \dot{S}&=&-\beta \frac{S}{N}I\\ \dot{I}&=&+\beta \frac{S}{N}I - \gamma I\\ \dot{R}&=&+\gamma I \end{eqnarray}

\(y=ax+b\)

A semi-mechanistic model for real-time forecasting

The unknown

  • Community/hospital/funeral transmission
  • Spatial dynamics
  • Changes in behaviour
  • Changes in reporting
  • Interventions
  • Seasonality
  • etc

The known

  • Average incubation period (~9 days)
  • Average infectious period (~11 days)
  • Case-fatality rate (~70%)

WHO Ebola response team (2014)

Time-varying transmission rate with smoothing prior

\(d\log \beta_t = \sigma dW_t\)

Dureau et al. (2013)

Particle MCMC

Use particle Markov chain Metropolis-Hastings to sample from \(p(\theta|\mathrm{Data})\) and \(p(\beta_t|\mathrm{Data})\)

Andrieu et al. (2010), Dureau et al. (2013), Murray et al. (2013)

  • \(d\log \beta_t = \sigma dW_t\)
  • Negative binomial observations, overdispersion \(\phi\)
  • \(\theta=\{\sigma, \phi, \beta_0, I_0\}\)
    • Intensity of random walk
    • Overdispersion of reporting
    • Initial transmission rate
    • Initial number infective

Camacho et al. (2015), Funk et al. (2016), Funk et al. (2017)

"We were losing ourselves in details […] all we needed to know is, are the number of cases rising, falling or levelling off?"

Hans Rosling

How good were the forecasts?

Comparison with null models

Calibration: Compatibility of forecasts and observations

Cumulative distribution of \(u_t=F_t(x_t)\)

Calibration: Compatibility of forecasts and observations

Divide \([0,1]\) in \(m\) subintervals

\(C({F_t},{x_t})=1-\frac{1}{2}\frac{m}{m-1}\sum_j \left|p_j - \frac{1}{m}\right|\)

\(p_j\) is the proportion of \(u_t=F_t(x_t)\) that is in interval \(j\)

Sharpness: Concentration of predictive distribution

\(S_t(F_t) = 1 - \frac{\mathrm{MADM}(y)}{m(y)}\)

\(y\) is a variable distributed according to \(F_t\)

\(\mathrm{MADM}(y)\): median absolute deviation about the median \(m(y)\) of \(y\)

"Evaluate predictive performance on the basis of maximising the sharpness of the predictive distribution subject to calibration"

Gneiting et al., J R Stat Soc B (2007)

Calibration: Compatibility of forecasts and observations.

Sharpness: Concentration of predictive distribution

Continuous ranked probability score

\(\mathrm{CRPS}(F_t,x_t) = E_{ F_t } \left|X - x_t\right| - \frac{1}{2}E_{F_t} \left|X - X'\right|\)

Quality of forecasts vs quality of decisions

Quality of forecasts vs quality of decisions

Quality of forecasts vs quality of decisions

Quality of forecasts vs quality of decisions

Evaluating quality of decisions

Outlook

Forecasts are becoming part of outbreak response

Forecasting challenges

Challenges in real-time modelling and forecasting

Need methods to
combine all available data streams
(individual/behavioural/spatial/genetic)

Challenges in real-time modelling and forecasting

Louis du Plessis, University of Oxford (unpublished)

Computationally efficient tools

Computationally efficient tools

Computationally efficient tools

Acknowledgements

Anton Camacho, John Edmunds, Roz Eggo,
Rachel Lowe, Adam Kucharski (LSHTM)
James Hensman (Lancaster), Lawrence Murray (Uppsala)

Thank you!

http://sbfnk.github.io